If P(-1,5) is hormonic conjugate of Q(-13,9) with respect to A and B. If A=(5,3) then find the length AB

Answer

To solve the problem, we need to find the length of segment AB given that P(-1, 5) is the harmonic conjugate of Q(-13, 9) with respect to points A(5, 3) and B.

### Step 1: Understand the Harmonic Conjugate Relationship
Since P and Q are harmonic conjugates with respect to A and B, the segments AB and AQ are in harmonic proportion. This means we can use the harmonic proportion relationship:
\[ \frac{2}{AB} = \frac{1}{AQ} + \frac{1}{AB} \]

### Step 2: Use the Distance Formula
To find the lengths AB and AQ, we will use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 3: Calculate Length AB
Let’s denote the coordinates of B as (x_B, y_B). We will first express AB in terms of B:
\[ AB = \sqrt{(x_B - 5)^2 + (y_B - 3)^2} \]

### Step 4: Calculate Length AQ
Next, we calculate AQ using point A(5, 3) and point Q(-13, 9):
\[ AQ = \sqrt{(-13 - 5)^2 + (9 - 3)^2} \]
Calculating the values:
\[ AQ = \sqrt{(-18)^2 + (6)^2} = \sqrt{324 + 36} = \sqrt{360} = 6\sqrt{10} \]

### Step 5: Substitute AQ into the Harmonic Proportion Formula
Now we substitute AQ into the harmonic proportion formula:
\[ \frac{2}{AB} = \frac{1}{6\sqrt{10}} + \frac{1}{AB} \]
Rearranging gives:
\[ \frac{2}{AB} - \frac{1}{AB} = \frac{1}{6\sqrt{10}} \]
This simplifies to:
\[ \frac{1}{AB} = \frac{1}{6\sqrt{10}} \]
Thus, we can find AB:
\[ AB = 6\sqrt{10} \]

### Step 6: Final Calculation
To find the exact length of AB, we can multiply:
\[ AB = 3\sqrt{40} = 3 \times 2\sqrt{10} = 6\sqrt{10} \]

### Conclusion
Therefore, the length of AB is:
\[ AB = 6\sqrt{10} \]